Chladni patterns for guitar plates

Chladni patterns show the geometry of the different types of vibration
of the guitar top plate. This site has an introductory explanation of modes
of vibration and a library of photographs of the Chladni patterns of a
guitar top plate and an intact guitar.

Modes of vibration

(See also the explanations of the guitar,
and Strings, harmonics and standing waves.)
A mode of vibration is just a way of vibration. Think what happens
when you strike a xylophone bar in the middle and set it vibrating. The
bar is supported at two points towards the ends. The simplest mode of vibration
is this: when the middle of the bar goes up (as shown by the solid lines
in the figure) the ends of the bar go down. When the middle goes down (dashed
lines), the ends go up. The two points that do not move are called nodes
and are marked N in the diagram. (If "modes" and "nodes" sound confusing,
remember that the node has no motion.)

Sketch of a simple
mode of vibration.

This first mode of the xylophone bar is rather similar to a mode of
vibration of a simple rectangular plate which is called the (0,2) mode
(the naming convention is explained below.)

Photographs of the Chladni pattern of:

mode (0,2) of a uniform rectangular aluminium plate.

In this pictures, the lines are formed from sand that has collected at
the nodes, but has been shaken off the moving regions. The top plate of
a guitar is more complicated in shape, and so the nodes also have a more
complicated shape. White sand was used for the black-painted aluminium
plate, and black sand for the guitar top plate.

Why are there nodes?

The supports of the xylophone bar do not cause the nodes, rather they are
placed at the positions which are nodes so as to facilitate this vibration.
In an object which is not firmly clampled, a vibration cannot easily move
the centre of mass of the object. It follows that, if some part is going
up, another part is going down. In the simple motion at resonance, the
point(s) that divide(s) these regions are nodes. When a violin or an isolated
part is vibrating, the centre of mass doesn't move much, so once again
it can be divided into parts that are going up and others that are going
down. In these simple modes of vibration, the motion of different parts
is either exactly in phase or exactly out of phase, and the two regions
are separated by nodes. The nodes are points for a quasi one-dimensional
object like a string, or lines for a quasi two-dimensional object like
a plate. (There is more explanation in Strings,
harmonics and standing waves.)

Modes of guitar plates

One of the modes is comparable with a mode of vibration of a rectangular
plate. In this mode the nodal lines separate the plate in three parts,
so it can be compared with the (0,2) mode of the rectangular plate, with
the middle part moving 180° out-of-phase with the ends.

How are Chladni patterns formed?

The plate can be made to resonate by a powerful sound wave which is tuned
to the frequency of the desired mode.

The plate can be bowed with a violin bow. This is easiest if one choses
a point that is a node for most of the modes that one doesn't want, but
not for the desired node.

The plate can be excited mechanically or electromechanically at the frequency
of the desired mode.

For the photographs on this site, a small (4g) magnet was fixed to the
bridge. An oscillating magnetic field (provided by a coil connected to
an audio amplifier and a signal generator) was used to provide an oscillating
force whose frequency is tuned to the resonance of the mode. Experiments
using different masses showed that the mass of the magnets caused us to
underestimate the frequency by about 10 Hertz in some cases. But there
were also patterns obtained with a magnet which could not be obtained without
a magnet (e.g. by a speaker). That means, that the vibrating system has
changed qualitatively by adding the magnet to the plate.

In all cases, some finely divided material is placed on the plate.
The material used here is fine sand. When the plate resonates, the motion
becomes large over most of the surface and this causes the sand to bounce
and to move about. Only at or near the node is the sand stationary. Thus
the sand is either bounced off the plate or else collects at the nodes,
as shown in the photographs.

Why are Chladni patterns useful?

The adjustment of the top plate is important to the properties of the final
instrument. The most important adjustments are thinning the wood towards
the edge of the plate, and thinning the braces. Chladni patterns provide
some feedback to the maker during the process of adjusting the plate to
its final shape. Symmetrical plates give symmetrical patterns; asymmetrical
ones in general do not.

It is very difficult to relate the frequencies of the modes of
the isolated top plate to those of the modes of a finished guitar. Fortunately,
adjustments can be made to an intact instrument: thinning the top plate
close to the edge, and adjusting the bracing by reaching through the soundhole
with specially shaped tools. The "tuning" of plate resonances is less formalised
in guitar making than in violin making. Over the centuries, violin makers
have discovered empirical relations between the modes of free plates and
the properties of the finished instrument. Many scientists have been interested
in the acoustics of violins, and many violin makers have been interested
in science, so a lot has been written about the acoustical properties of
violins and their parts. See: